In the diagram, HJLM is a square, and GH = 10. Find the area of trape

Question

gpp-sgf_img3.png In the diagram, HJLM is a square, and GH = 10. Find the area of trapezoid GHJK. A. 50+25\sqrt{3} B. 50+50\sqrt{3} C. 75+12.5\sqrt{3} D. 75+25\sqrt{3} E. 75+50\sqrt{3} Kudos for a correct solution .

littlegirl · Answer

Answer is D
Area of GHJK = 2 * Area of GHM+ Area of HJLM

= 2* (GM* HM)/2 + HM^2

Since GHM is special right triangle we have GM= GH/2=5 ; HM=GM *sqrt(3)=5*sqrt(3)
then Area =2 * 5*5*sqrt(3)/2+ (5*sqrt (3))^2
= 75+25*sqrt(3)

aj13783 · Answer

If you split the trapezium, you get a square and 2 right angled triangles

In the Right angled triangle, Hypotenuse = 10 and angle = 60 degree

Opposite side/hypotenuse = sin 60

opp side/10 = sq.rt(3)/2

opp side = 5sq.rt(3)

Thus Base = 5

Area = 2(5*5*sq.rt(3))/2+ (5sq.rt(3))^2

= 75+25sq.rt(3)

Option D

PareshGmat · Answer

Answer = D. 75+25\sqrt{3} Refer diagram below: gpp-sgf_img3.png Sides of right triangle 30:60:90 are in the ratio 1: \sqrt{3}:2 So the dimensions of the sides would be as shown in the diagram Area of trapezoid = \frac{1}{2} (5 + 5 + 5\sqrt{3} + 5\sqrt{3}) * 5\sqrt{3} = 75+25\sqrt{3}

Bunuel · Answer

MAGOOSH OFFICIAL SOLUTION:

This entire problem hinges on recognizing that triangles GHM and JKL are 30-60-90 triangles and using the properties of those triangles. First of all, notice that because HJLM is a square, it must be true that HM = JL. This means, the two triangles, GHM and JKL, must be congruent and have all the same sides & angles. GH = JK = 10.

Now, in a 30-60-90 triangle, the smaller leg is half the hypotenuse, so GM = LK = 5. The longer leg is the square root of 3 times the shorter leg, so

HM = JL = 5\sqrt{3}

This is the side of the square, so we square this to get the area of square HJLM.

Area \ of \ HJLM = (5\sqrt{3})^2=75

That’s part of the area. Now, we need the area of the triangles. One triangle is A = 0.5bh, so two triangles would simply be A = bh

Area \ of \ triangle=bh =GM*HM=5*5\sqrt{3}=25\sqrt{3}

Total area = square + triangles =  75 + 25\sqrt{3} .

Answer = (D)

bumpbot · Answer

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In the diagram, HJLM is a square, and GH = 10. Find the area of trape

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In the diagram, HJLM is a square, and GH = 10. Find the area of trape

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